exchange algebras, differential galois theory and...

Exchange algebras,differential Galois theoryand Poisson Lie groups

Michael Semenov-Tian-Shansky

Institut Mathematique de Bourgogne, Dijon, France,

and Steklov Mathematical Institute, St. Petersburg, Russia

Based on joint work with Ian Marshall

Exchange algebras,differential Galois theoryand Poisson Lie groups – p. 1/5

The Setting

The space H of Schroedinger operators on the circle

H = −∂2x − u, u ∈ C∞(S1), S1 � R/2πZ,

is the phase space for the KdV hierarchy (with periodicboundary conditions). It carries a family of naturalPoisson structures which play an important rôle in theHamiltonian description of the KdV flows.

The Virasoro Poisson operator:

l = 12∂

3x + u∂x + ∂xu.

Problem:Extend the Poisson structure to the space of wavefunctions.


The tower of KdV-like equations

A closely related practical question: The tower ofKdV-like equations.


The tower of KdV-like equations–2

As we shall see, this diagram is naturally understood interms of the differential Galois theory.This is the point of view suggested by George Wilson in1989.

Problem: equip all phase spaces in this diagram withnatural Poisson structures in such a way that all arrowsbecome Poisson mappings.


General problem

In a more abstract language, second order differentialoperators on the line are associated with projectiveconnections; solutions of the Schroedinger equationsare covariantly constant sections of the associatedprojective bundle.In a similar way, the space of flat linear connections ona circle carries a natural Poisson structure. Thequestion is:

Extend this Poisson structure to the space ofcovariantly constant sections.In both cases, obstructions are connected withnontrivial cohomology:

In linear case, it is the Maurer–Cartan cocycle.In projective case it is the Gelfand–Fuchs cocycle.


Elementary theory

The space V = Vu of solutions of the Schroedingerequation

−ψ′′ − uψ = 0

is 2-dimensional.

Any two solutions φ, ψ have constant wronskianW = φψ′ − φ′ψ.

An element w ∈ V may be regarded as a quasi-periodicplane curve (such that w ∧ w′ is nowhere zero).

Monodromy matrix:

w(x+2πn) = w(x)Mn, n ∈ Z, wherew = (φ, ψ) is a row vector

The group G = SL(2) acts naturally on V (preservingthe wronskian) by right multiplication.


Classical Theorem:

Any pair of linearly independent solutions of theSchroedinger equation defines a non-degeneratequasi-periodic projective curve γ : R → CP1 such that

γ(x+ 2π) = γ(x)M.

Conversely, any non-degenerate quasi-periodicprojective curve may be lifted to a non-degeneratecurve in C

2 such that its wronskian is equal to 1 andhence gives rise to a 2nd order differential equation.


Schwarzian Derivative

Fix an affine coordinate on CP1 in such a way that ∞corresponds to the zeros of the second coordinate ψ of thepoint on the plane curve; with this choice γ is replaced withthe affine curve x �→ η(x) = φ(x)/ψ(x). The potential u maybe restored from η by the formula

u = 12S(η), where S(η) =

η′′′

η′− 3

2

(η′′

η′

)2

is the Schwarzian derivative.

S(u) is projective invariant: if η = aη+cbη+d , then S(η) = S(u).


Point of view of differential Galois theory

We define the differential field C〈ψ1, ψ2〉 as a freealgebra of rational functions in an infinite set ofvariables ψ1, ψ2, ψ

′1, ψ

′2, ψ

′′1 , ψ

′′2 , . . . with a formal

derivation ∂ such that ∂ψ(n)i = ψ

(n+1)i .

A differential automorphism is an automorphism ofC〈ψ1, ψ2〉 (as an algebra) which commutes with ∂. Alldifferential automorphisms are induced by lineartransformations (ψ1, ψ2) �→ (ψ1, ψ2) · g, g ∈ GL(2,C).

Let (W ) be the differential ideal in C〈ψ1, ψ2〉 generatedby ψ1ψ

′2 − ψ′

1ψ2 − 1. Automorphisms which preserve Wbelong to G = SL(2).

The differential subfield of G-invariants coincides withC〈u〉.


Intermediate differential fields

Let Z = {±1} be the center of G and N , A, B = AN itsstandard subgroups (nilpotent, split Cartan & Borel). Thesubfields of invariants are freely generated differentialalgebras:

C〈φ, ψ〉Z = C〈η〉, η = φ/ψ,

C〈η〉A = C〈ρ〉, ρ = η′/η,

C〈η〉N = C〈θ〉, θ = η′,

C〈η〉B = C〈v〉, v = η′′/η′,

C〈η〉G = C〈u〉, u = S(η).


Extension Tower

C〈η〉

C〈η〉A� �

��

C〈η〉N��

��

C〈η〉B� �

��

��

��

C〈η〉G��

��


KdV–like equations once again

Basic classical observation: there is a family ofcompatible KdV-like flows on each level of the towerrelated by natural differential substitutions:


Main Problem:

Equip all levels of the tower with Poisson structure so asto make all arrows Poisson mappings.This is a surprisingly non-trivial question.

Attempt of G.Wilson: Look at symplectic forms whichcan be pulled back.Obstructions:

When monodromy is nontrivial, symplectic formsfail to be closed!

Our approach:Guess the answer from natural covariancerequirements.


Main result:

The Poisson bracket on the (projectivized) space ofwave functions is rigid and essentially unique; theGalois group automatically becomes a Poisson groupwith the standard Poisson structure; other possiblePoisson structures on G are excluded.


A reminder on Poisson Lie groups

Let G be a Lie group with Lie algebra g. A Poisson structureon G is called multiplicative if the multiplication

m : G×G→ G

is a Poisson mapping. A Lie group equipped with amultiplicative Poisson bracket is called a Poisson Lie group.An action G×M → M of a Poisson group on a Poissonmanifold M is called a Poisson action if this mapping isPoisson; in other words, for F,H ∈ Fun(M), their Poissonbracket at the transformed point g ·m ∈ M may becomputed as follows:

{F,H}M (g ·m) ={F (m, · ), H(m, · )

}G

(g) +{F (· , g), H(· , g)

}M


In that case we shall also say that the Poisson bracket onM is G-covariant.

Important fact:Multiplicative Poisson bracket on G gives rise to thestructure of a Lie algebra on the dual space g∗;The dual of the commutator map [ , ] : g∗ × g∗ → g∗ isa 1-cocycle on g.

A pair (g, g∗) with these properties is called a Liebialgebra.

Fundamental Theorem (Drinfeld):Multiplicative Poisson bracket is completelydetermined by its linearization.


Hence there is an equivalence of two categories:

Category of Poisson Lie groups (morphisms = Liegroup homomorphisms which are also Poissonmappings)

Category of Lie bialgebras (morphisms =homomorphisms of Lie algebras such that their dualsare homomorphisms of the dual algebras).

Important tool:Restrict Poisson action to subgroups.

Obvious possibility:· Restrict to Poisson subgroups.More flexible option:· Restrict to admissible subgroups.


Admissible subgroups and Poisson Reductio

Definition:A subgroup H ⊂ G of a Poisson Lie group G is calledadmissible if the subalgebra of H-invariantsFun(M)H ⊂ Fun(M) is closed with respect to thePoisson bracket.

Admissibility criterion:

H ⊂ G is admissible if and only if h⊥ ⊂ g∗ is a Liesubalgebra;

H ⊂ G is a Poisson subgroup if and only if h⊥ is anideal in g∗.

Poisson Reduction:If H is admissible, Fun(M)H � Fun(M/H) inheritsthe natural Poisson structure.


The case of SL(2)

The group G = SL(2,C) carries a family of naturalPoisson structures called the Sklyanin brackets whichmake it a Poisson Lie group.

These Poisson structures are parameterized by thechoice of a classical r-matrix r ∈ g ∧ g.In usual tensor notation we have

{g1, g2} = [r, g1 g2], (1)

Important fact:Up to natural equivalence there exist three types ofclassical r-matrices:

(a) r = 0;(b) r = h ∧ f(c) r = ε e ∧ f , where ε is a scaling parameter.


Case (a) gives trivial bracket; case (b) (the so calledtriangular r-matrix) is degenerate. Case (c) (alias,quasitriangular or factorizable case) is generic.

Explicit formulae in case (c):

Let g =(

α βγ δ

). Then

{α, β} = εαβ, {α, γ} = εαγ,

{β, δ} = εβδ, {γ, δ} = εγδ,

{β, γ} = 0, {α, δ} = 2εβγ.

Important:det g = αδ − βγ is a Casimir function and hence thePoisson bracket is well defined on the coordinatering of SL(2) and even of PSL(2).)


Yang-Baxter tensor

[[r, r]] = [r12, r13] + [r12, r23] + [r13, r23] ∈ ∧3g

This tensor naturally arises in the check of the Jacobiidentity.

Important to notice:For g = sl(2) the Yang–Baxter equation does notimpose any restrictions on r, since ∧3g � C is trivial.

However, there are still two different cases todistinguish.

[r, r] = 0 in cases (a) and (b).

[r, r] = −ε2 �= 0 in case (c) (r standard quasitriangularr-matrix).


The Dual Group

The dual Lie algebra g∗ associated with the standardr-matrix is

g∗ = {(X+, X−) ∈ b+ ⊕ b− | diagX+ + diagX− = 0} .The dual Lie group is

G∗ = {(b+, b−) ∈ B+ × B−|diag b+ · diag b− = I} .Important fact:

The dual group is also a Poisson Lie group.

The Poisson bracket on G∗ may be described in termsof Poisson bracket relations for matrix coefficients of(b+, b−). There is another way based on the Gaussdecomposition mapping G∗ → G(b+, b−) �→ b+b

−1− :


The dual Poisson structure

Important assertion:The mapping

G∗ → G : (b+, b−) �→M = b+b−1−

maps G∗ onto an open dense subset in G; theinduced Poisson structure extends smoothly to theentire manifold G.Explicitly we have

{M1,M2} = M1M2r + rM1M2 −M2r+M1 −M1r−M2,(2)

where r± = r ± εt and t ∈ g ⊗ g is the tensor Casimirelement.


Back to the study of wave functions:

Starting point: The space W of all quasi-periodic planecurves,

W = {(w = (φ, ψ),M) | w(x+ 2π) = w(x)M} .Scaling group: C := C∞(R/2πZ,R×).

Constraint set:

W ′ = {w ∈ W;W (w) = 1} .W ′ is the cross-section of the scaling action; it may beidentified with the space V of wave functions.Hence V has a twofold description:

as a quotient spaceas a subspace W ′ ⊂ W.


Covariance and exchange bracket

Another action on W: the natural action of G = SL(2).

Covariance axiom:C ×W → W and G×W → W are Poisson maps.This presumes that both C and G may carry Poissonbracket (possibly trivial) which makes them Poisson Liegroups; their action is then Poisson action.

Lemma 1.Suppose that the bracket on W is covariant withrespect to the action of C. Then the Poissonstructure on C is trivial and, writing w = (φ, ψ), thebracket of evaluation functionals has the form:

{w1(x), w2(y)} = w1(x)w2(y)R(x, y), (3)


Exchange matrix

The “exchange matrix” R(x, y) ∈ Mat(4) is given by

R(x, y) =

⎛⎜⎜⎜⎜⎜⎝

A(x − y) 0 0 0

0 B(x − y) −C(y − x) 0

0 C(x − y) −B(y − x) 0

0 0 0 D(x − y)

⎞⎟⎟⎟⎟⎟⎠

.

Let us drop temporarily the Jacobi identity and considerall Poisson brackets of this type which are G-covariant.

First the case of G-invariant brackets:


Jacobi Identity: case of trivial bracket on G

Lemma 2.Assume that the Poisson bracket (3) isright-G-invariant (hence G carries trivial bracket);then the exchange matrix has the structure

R0(x, y) = a(x− y)I +

⎛⎜⎜⎜⎝

0 0 0 0

0 c(x− y) −c(x− y) 0

0 c(x− y) −c(x− y) 0

0 0 0 0

⎞⎟⎟⎟⎠ ,

where a and c are arbitrary odd functions.


Jacobi Identity: nontrivial case

Now assume that G carries a nontrivial Poisson structure.

Lemma 3.Assume that the Poisson bracket (3) isright-G-covariant; then the exchange matrix has thestructure

Rr(x, y) = R0(x, y) + r, (4)

Recall the classification of r-matrices: either(1) [[r, r]] = 0 (r = 0 or r degenerate, cases (a)& (b);

(2) or [[r, r]] = −ε2 �= 0 (r quasitriangular ).

In case (2) the Yang–Baxter tensor [[r, r]] gives an extraterm to the Jacobi identity for the exchange bracket.


Lemma 4.The exchange bracket (3) with exchange matrix (4)satisfies the Jacobi identity if and only if

c(x − y)c(y − z) + c(y − z)c(z − x) + c(z − x)c(x − y) = 0

in case (1) and

c(x − y)c(y − z) + c(y − z)c(z − x) + c(z − x)c(x − y) = −ε2

in case (2).


Solution of the functional equation

Solution:cλ(x− y) = ε cothλ(x− y) in case (2).

Important limiting case λ→ ∞: c(x− y) = ε sign(x− y).

In case(1) the solution is c(x− y) = λx−y .

Poisson bracket relations for the projectivized wavefunctions:

{η(x), η(y)} = ε(η(x)2 − η(y)2

) − c(x− y) (η(x) − η(y))2 .(5)

Proposition.Formula (5) defines a family of G-covariant Poissonbrackets on the space of “projective curves” η.Notice that a drops out, but there is still somefreedom in the choice of c.


Wronskian constraint

One more key ingredient:the wronskian constraint.

It fixes c completely and allows to choose a in a natural wayas well.

We have

{W (x), φ(y)} = (c(x− y) − 2a(x, y))W (x)φ(y)

− c′(x− y)φ(x)[φ(x)ψ(y) − ψ(x)φ(y)]. (6)

A similar formula holds for {W (x), ψ(y)}.


Crucial observation

Proposition.The constraint W = 1 is compatible with the Poissonbrackets for the scaling invariant η if and only if thelast term in (6) is identically zero; this is possible ifand only if c′(x− y) is a multiple of δ(x− y), i.e., ifc(x− y) is a multiple of sign(x− y).

Without restricting the generality, we can now assumethat r is the standard quasitriangular r-matrix, r = e ∧ f .Other possible choices differ by rescaling andconjugation.

Standard fact:With this choice, B, A, N , together with the oppositeBorel subgroup, give the complete list of admissiblesubgroups of G.


Spontaneous symmetry breaking

Important conclusion:Differential Galois group spontaneously becomes aPoisson group and the choice of its Poissonstructure is essentially unique.


Poisson bracket relations for the wronskian

We have:

{W (x),W (y)} = (sign(x− y) − 2a(x− y))W (x)W (y), (7)

or, equivalently

{logW (x), logW (y)} = (sign(x− y) − 2a(x− y)). (8)

Proposition.Assume that a is so chosen that

sign(x− y) − 2a(x− y) = δ′(x− y).

(In other words, a(x− y) is the distribution kernel ofthe operator 1

2

(∂−1 − ∂

).) Then:


Wronskian as the moment map

(i) The logarithms of wronskians form a Heisenberg Liealgebra, the central extension of the abelian Lie algebraof C.

(ii) Let C′ = C/C∗ be the quotient of the scaling group overthe subgroup of constants; logW is the moment map forthe action of C′ on W.

The resulting picture:The scaling action is Hamiltonian with moment mapμ = logW .V arises as a result of Hamiltonian reduction withrespect to C over the zero level of μ.The constraint set logW = 0 is (almost)non-degenerate (i.e., this is a 2nd class constraint,according to Dirac).


Poisson brackets for the monodromy

Remark.The projective invariants η(x) commute with thewronskian and hence their Poisson brackets are notaffected by the constraint.

Theorem.

{w(x)1,M2} = w(x)1

[M2r+ − r−M2

],

{M1,M2} = M1M2r + rM1M2 −M2r+M1 −M1r−M2.

(9)

The Poisson bracket for the monodromy is precisely thePoisson bracket of the dual Poisson group G∗. In otherwords:


Monodromy as a non-abelian moment map

Theorem.The ‘forgetting map’ μ : (w,M) �→M is a Poissonmorphism from W into the dual group G∗.

AssertionThe mapping μ is the non-abelian moment mapassociated with the right action of G on W

Reminder:A nonabelian moment map associated with aPoisson group action G×M → M is a mapping tothe dual Poisson group, μ : M → G∗; in our case, μis simply described if the dual group is modeled on Gvia the Gauss factorization map.


Basic Poisson bracket relations

Basic Poisson bracket relations for differential Galoisinvariants:

{η(x), η(y)} = η(x)2 − η(y)2− sign(x− y) (η(x) − η(y))2.For θ = η′ we have

{θ(x), θ(y)} = 2 sign(x− y)θ(x)θ(y).

For v = 12η

′′/η′ = 12θ

′/θ we have

{v(x), v(y)} = 12δ

′(x− y).

For u = 12v

′ − v2 = S(η) we have

{u(x), u(y)} = 12δ

′′′(x− y)+ δ′(x− y)[u(x)+u(y)

]. (10)


Nonlocal bracket for A-invariants

The last formula (10) reproduces the Virasoro bracket(which has not been assumed a priori !)

Poisson bracket relations (38) – (10) listed above arealgebraic. Since the basic Poisson bracket relations (5)are nonlocal, this need not always be the case. This iswhat happens in the case of A-invariants:

The differential subalgebra of A-invariants in C〈η〉 isgenerated by ρ = η′/η.The Poisson brackets for ρ have the form

{ρ(x), ρ(y)} =

2ρ(x)ρ(y)

[sinh

∫ y

xρ(s) ds+ sign(x− y) cosh

∫ y

xρ(s) ds

].


Back to the extension tower

TheoremAll arrows in the extension tower

C〈η〉

C〈η〉A� �

��

C〈η〉N��

��

C〈η〉B� �

��

��

��

C〈η〉G��

��

are Poisson algebra homomorphisms.


Back to the extension tower–2

TheoremAll arrows in the commutative diagram

are Poisson mappings.Exchange algebras,differential Galois theoryand Poisson Lie groups – p. 41/5

Theorem (second part)The KdV-like equations listed in the diagram aregenerated by the standard Hamiltonian

H =

∫u2 dx.

(which lives on the bottom level of the extensiontower but can be pulled back to all upper levels)A similar assertion holds for all higher KdV flows.All flows factorize over those lying beneath.


Discrete case

Second order difference equations on theone-dimensional lattice with periodic potential

φn+2 + unφn+1 + φn = 0, un+N = un. (11)

In operator form(τ2 + u τ + 1

)φ = 0, (12)

where τ is the shift operator, (τφ)n = φn+1.

Elementary Theory:For a given u, the space of its solutions istwo-dimensional;Any two solutions φ, ψ have constant wronskianW = φnψn−1 − φn−1ψn.


Projective picture

Projective description:An (ordered) projective configuration is a mapγ : Z → CP1.A configuration is non-degenerate if γn �= γn+1. for alln.A plane configuration is a map w : Z → C

2; it isnon-degenerate if wn ∧ wn+1 �= 0.We denote wn by the row vector (φn, ψn).

Theorem.Any pair of independent solutions of the discreteSchroedinger equation defines a non-degeneratequasi-periodic projective configuration γ : Z → CP1

such that γn+N = γn ·M , where M is the monodromy.


Theorem (suite).Any two projective configurations associated with agiven discrete Schroedinger equation are related bya global projective transformation.Any non-degenerate quasi-periodic projectiveconfiguration may be lifted to a non-degenerateplane configuration such that its wronskian

W [w]n := φnψn−1 − ψnφn−1 = 1.

We replace the projective line with its affine modelputting ηn = φn/ψn. The group G = SL(2) is the(difference) Galois group of equation (11).

G acts on ηn by fractional linear transformations.


Schwarzian vs cross-ratio

The obvious analogue of the Schwarzian is thecross-ratio

sn[η] := [ηn, ηn+1, ηn+2, ηn+3] =ηn − ηn+2

ηn − ηn+1· ηn+1 − ηn+3

ηn+2 − ηn+3;

Curious fact:The potential u cannot be directly restored from η.Instead, we have:sn = unun+1.

If the lattice length is odd, the potential still may beuniquely restored from sn by solving a quadraticequation. Hence in this case C〈u〉 is a quadraticextension of C〈η〉G.


Covariant Poisson brackets

Denote by W the space of all plane quasi-periodicconfigurations and by C the discrete scaling group.

Proposition.Assume that the Poisson structure on W is covariantwith respect to the right action of G and to the naturalaction of the scaling group. Then it has the form

{w1

m(x), w2n(y)

}= w1

m(x)w2n(y)R(m− n),

where

R(k) = R0(k) + r, R0(k) = akI +

⎛⎜⎜⎜⎝

0 0 0 0

0 ck −ck 0

0 ck −ck 0

0 0 0 0

⎞⎟⎟⎟⎠ ,


Proposition (suite).ak is an arbitrary odd function;ck is an odd function which satisfies

cn−mcm−k + cm−kck−n + ck−ncn−m = α, (13)

where α = 0 when r = 0 or r triangular and α = −ε2for r quasitriangular. (In the sequel, we set ε = 1).

The space V ⊂ W of discrete wave functions is definedby the constraint W [w] = 1.

Poisson bracket relations for the Wronskian:

{Wn, φm} = (an−m + an−1−m − cn−m)Wnφm

+ (cn−m − cn−m−1)(φnφn−1ψm − φnψn−1φm). (14)


Proposition.Scaling invariants ηn commute with the wronskian ifand only if the second term in (14) is proportional toWnφm; hence

cn−m − cn−m−1 = (δnm + δn,m+1). (15)

Important fact:This recurrence relation is again solved by the signfunction.

Hence we get:

{ηn, ηm} = η2n − η2

m − sign(n−m)(ηn − ηm

)2. (16)


Remark.This is in fact a Poisson subalgebra of thecontinuous algebra.Of course, it is not true that the wave functions of thediscrete equation are the values of the wavefunctions for the continuous equation!

Poisson bracket relations for Galois invariants.We have:

C(η)N = C(θ), where θm := ηm+1 − ηm

C(η)B = C(λ), where

λm :=ηm+2 − ηm+1

ηm+1 − ηm·


Poisson bracket relations

We get: {θm, θn} = −2 sign(m− n)θmθn,{λm, λn} = 2

(δm+1,n − δm,n+1

)λmλn.

A natural interpretation of the variables λn is connectedwith the Miura transform for the discrete Schroedingerequation.

Assume that the difference operator (12) is factorized,

τ2 + u τ + 1 = (τ + v)(τ + v−1). (17)

The potentials u, v are related by the differenceMiura map,

un = vn + v−1n+1. (18)


Poisson bracket relations–2

Assume that ψ satisfies the first order equation(τ + v−1)ψ = 0. Let φ be the second solution of thisequation such that W (φ, ψ) = 1 and η = φ/ψ; then

ηn+1 − ηn =1

ψnψn+1.

Clearly, vn = −ψn/ψn+1 and hence

vnvn+1 =ψn

ψn+2=

ψn+1ψn

ψn+2ψn+1=ηn+2 − ηn+1

ηn+1 − ηn= λn (19)

Thus λn is the product of two neighbouring potentials inthe factorized Schroedinger operator.


Poisson bracket relations–3

Remark.The potentials vn again are not rational Galoisinvariants of B and belong to a quadratic extensionof C(λ).

We have:

sn = unun+1 =(1 + λn)(1 + λn+1)

λn+1. (20)

Poisson bracket relations for λn and sn:

{λm, λn} = (δm+1,n − δm,n+1)λmλn,

{sm, sn} =(δm+1,n − δm,n+1

)(sm + sn − smsn)

+ smsn(s−1m+1δm+2,n − s−1

n+1δm,n+2

).

(21)


Discrete Virasoro algebra

Proposition.Let Φn = (−1)n signn, n �= 0, Φ0 = 0. Then

{vn, vm} = 2Φn−mvnvm, (22)

{un, um} = 2Φn−munum + 2(δm+1,n − δm,n+1). (23)

RemarkFormula (22) coincides with the lattice Virasoroalgebra introduced by Frenkel, Reshetikin & myself,while (21) coincides with the Faddeev–Takhtajanversion of the lattice Virasoro algebra. Structureconstants Φn−m arise in paper of Frenkel, Reshetikin& myself in the framework of the discreteDrinfeld–Sokolov theory.


Possible Extensions:

1. Energy dependent case;

2. q-difference case;

3. Higher order equations.


exchange algebras, differential galois theory and...

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